Search

The article presents a response to the paper "Observations on Jensen's Experimental Design for Study of Effects of Accounting Variations in Decision Making" by Thomas R. Dyckman and discusses the scale of observation in an analysis of variance (ANOV) model. Dyckman questions Jensen's research in proposing violations in the underlying assumptions of the ANOV model and whether or not these violations would appreciably affect the conclusions of his experiment. An interval scale was used during some of Jensen's data selection for discussion which Dyckman feels is incorrect in its additivity assumption.

This article analyzes the second-order interaction in multidimensional contingency tables in marketing research. THe author examines the introduction of chi-square analysis in marketing research as well as the use of chi-square to study three-dimensional contingency tables. The author also examines the association of the three attributes of multidimensional contingency tables. The author suggests that these complex techniques are needed to examine the more complicated relationships that exist between marketing and consumers.

The article discusses statistical testing methods of comparing advertisements and their overall effectiveness. Worthwhile measuring instruments should satisfy three basic requirements. The first is reliability. Its repeated application under essentially similar circumstances should produce essentially the same results. The second is validity. It should measure what it is supposed to measure. The third is sensitivity. A crude, and not very satisfactory general statement is that portfolio tests should produce a "statistically significant" difference in favor of the better ad most of the time when an important difference exists in the impact of two test ads.

Monte Carlo methods were used to study the performance of the largest-root test, two trace-type tests and three determinantal tests in the fixed-effects MANOVA model when certain assumptions are violated. Results indicated that for protection against nonnormality and heterogeneity of covariance matrices, the largest-root test should be avoided, while the Pillai-Bartlett trace test may be recommended as the most robust of the MANOVA tests, with adequate power to detect true differences in a variety of situations. [ABSTRACT FROM AUTHOR]

Exact small sample behavior in two-way contingency tables is investigated for Pearson's chi-square statistic (X²), Light and Margolin's C statistic and its related R² measure of association, Kuliback's minimum discrimination information statistic (2&lcirc;), and Goodman and Kruskal's Lambda. R² is shown to be identical to Goodman and Kruskal's tb, leading to a test for independence based on 4. In small samples from a product of multinomials model, the null distribution of C is better approximated by a x² distribution than is the null distribution of X²; both are considerably better approximated by a x² distribution than is the null distribution of 2&lcirc;. It is proved for tables with two columns and any number of rows that if the column totals are equal, then X² ≤ 2&lcirc;; thus, X² is more conservative than 2&lcirc;. Hence, use of 2&lcirc; should be avoided in testing independence in tables with small samples. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions for admissibility are given for test procedures based on a preliminary test of significance. Three types of problems are studied-testing the normal mean, fixed effects models of the analysis of variance and random effects models. Admissibility in this instance is equivalent to the intuitive and practical condition that acceptance regions of the procedures have convex sections in certain variables, while other variables are fixed, It is easy to check when the conditions hold. A discussion of optimality properties of these and other types of pooling procedures is given. [ABSTRACT FROM AUTHOR]

Three statistics for testing additivity and making detailed inferences on interactions in a two-way setup are considered. The first method uses the well-known sum of squares for testing and yields Scheffe bounds for all contrasts; the second introduces a maximum characteristic root statistic and Roy bounds, but is restricted to product-type contrasts; the last is limited to tetrad differences and its statistic is the maximum of all such differences. Bounds on percentage points for this maximum are provided in the appendix. Uses of the three methods in inferring various types of interaction contrasts are discussed and illustrated. [ABSTRACT FROM AUTHOR]

Mean square errors associated with median response estimates are compared for the normal and logistic distributions, based on both maximum likelihood and minimum chi square analyses. In broad perspective there is little difference among the mean square errors for these four estimates, regardless of sample size and stimulus level spacing. The first three of the median response estimates require an iterative numerical solution, whereas the minimum chi square analysis for the logistic distribution yields an explicit expression for the associated median response estimate. This explicit estimation expression may be particularly convenient when nonuniform spacing of stimulus levels is involved in testing. [ABSTRACT FROM AUTHOR]

A test for the hypothesis of zero second-order interaction is proposed. The test is obtained by making use of conditional reference sets which are derived in Zelen [17]. This permits the derivation of an exact conditional test of significance for testing the hypothesis of zero second order interaction. The techniques used, in approximating the large sample distribution of this test, are extensions of those due to Gart [5] and Zelen [17]. The test is applied to a numerical example. [ABSTRACT FROM AUTHOR]

Hypothesis test procedures for tests of separate families of hypotheses are defined. These procedures are then compared using Monte Carlo samples as data for five pairs of invariant distributions. The comparisons are made by calculating an approximate relative efficiency for each procedure with respect to the likelihood ratio procedure, which for three pairs of distributions considered is uniformly most powerful among the class of invariant procedures, [ABSTRACT FROM AUTHOR]

The problem considered is how to compare time-to-response data for different groups when the group membership of an individual can be arbitrarily varied during a study. Modified life tables can be constructed which reflect such changes in an individual's status, and associated measures of relative risk and statistical significance calculated. This is illustrated with survival data for heart-transplant patients, for which a patient can transfer from the nontransplanted to the transplanted group. Alternative procedures are given in which distinctive groups are defined for each transplant day. [ABSTRACT FROM AUTHOR]

The asymptotic method described in an earlier article (Bowman and Shenton [2]) is applied to derive the power of the square root of b[sub 1] test of normality against skewed alternatives. In particular, normal mixtures of two components and Type A distributions, each depending on four parameters, are considered. Comparisons of the assessment of percentage points of square root of b[sub 1] using Cornish-Fisher expansions based on the normal and Monte Carlo simulations are given. [ABSTRACT FROM AUTHOR]

Cochran's Q-test, proposed to test the equality of three proportions for correlated observations, may have a larger asymptotic significance level than the nominal one, unless the admissible family of distributions is restricted to ensure that under the hypothesis the correlations between the observations are also equal. For a large class of such restricted families, optimal C[sup 2](alpha) tests of the hypothesis are obtained, and the family for which the Q-test is an optimal C[sub 2](alpha) test is identified. The use of Q for testing a related hypothesis, suggested by Madansky, is also discussed: it is shown that the Q-test is not consistent against all alternatives to Madansky's hypothesis, while the classical chi[sup 2]-test provides an optimal C[sub 4](alpha) test for it. [ABSTRACT FROM AUTHOR]

Procedures for discriminating between models from separate families of hypotheses are examined, with principal emphasis on procedures invariant under location and scale transformations. The discrimination procedures considered are compared using Monte Carlo samples as data for five pairs of invariant distributions. These comparisons are made with respect to the best invariant procedure, a procedure requiring no knowledge of sampling distributions, using approximate relative efficiencies calculated from the Monte Carlo results. On the basis of these efficiencies and the computational complexities of the procedures, suggestions are made for their use. [ABSTRACT FROM AUTHOR]

Procedures are given for testing hypotheses and obtaining confidence intervals for the reliability of systems of k independent parallel components consisting of m < k component types. Assume that there are alpha[sub I] components of the ith component type, I = 1, 2 ..... m and that p, is the failure probability of the ith component type. Then the probability of failure of the system is rho = II[sup m, sub l=1] p[sup alpha, sub l]. If m independent sequences of Bernoulli trials are conducted, one for each component type, techniques which may be employed for statistical inference concerning p are given. These techniques are suitable under those conditions for which the binomial distribution may be satisfactorily approximated by the Poisson distribution. [ABSTRACT FROM AUTHOR]

The asymptotic non-null distribution is obtained for the modified form of the Pearson chi-square statistic studied by Dahiya and Gurland [3]. By utilizing this result the power is obtained for specific alternative distributions in testing for normality, This enables recommendations to be made as to the number of class intervals to be employed in applying the aforementioned modification of the Pearson chi-square test of normality. [ABSTRACT FROM AUTHOR]

We derive an asymptotically efficient method of estimating the coefficients of systems of seemingly unrelated regression equations when a first-order vector autoregressive error specification is hypothesized. This error specification is shown to contain several commonly used specifications as special cases. A Monte Carlo experiment was performed to examine the finite sample performance of our estimator in comparison with several other estimators. The general results were that the new estimator performed relatively well. [ABSTRACT FROM AUTHOR]

The present model is applicable when pi[sub Y], the population proportion with a nonsensitive attribute Y, is not known in advance and two samples are required to estimate pi[sub A], the population proportion with a sensitive attribute A. It consists of using the sensitive question and two nonsensitive alternate questions in each sample, yielding two unbiased estimates of pi[sub A]. The optimum estimator, a weighted average of these two estimates, is described and its minimum variance calculated. The efficiency of this estimator is compared with Moors' optimized version of the standard two sample single alternate question model, and with the one sample single alternate question model when pi[sub Y] is known. [ABSTRACT FROM AUTHOR]

If the research goal in applying classical discriminant analysis techniques is to test for differences in the general characteristics of groups, then standard tests from multivariate analysis of variance are appropriate. Statistically significant differences among group means and dispersions indicate that the groups arose from separate populations. If, however, the goal is to explore group structure in terms of group overlap and the locations of individual observations relative to each other and to the group means, then it can be shown that the significance tests provide incomplete and often misleading representations of the data. Similarly, in problems involving just more that just two groups, the null hypothesis can be rejected, when in fact only one of the several groups arose from a different population.

This article analyzes the relationship between stochastic dominance (SD) and mean-semivariance (ES) criteria. This paper contains two main results. First, there are good conceptual bases for anticipating greater consistency between ES and second SD (SSD) rules than between mean-variance (EV) and SSD rules. In particular, with one possible and relatively minor exception (identical mean and semivariance, the efficient set of portfolios derived from the ES rule with semivariance calculated around a fixed point is a subset of the SSD efficient set.

This article presents an analysis of variance techniques for unbalanced data. Some of the more important issues encountered in the analysis of unbalanced data is examined here within a single framework. These issues are broadly defined as solutions to the normal equations for linear models where the incidence or design matrix is not of full column rank, estimable functions of the parameters and tests of hypotheses, and reductions in sums of squares and interpretation of results.

An F-test of linear hypotheses is compared with Bonferroni t-tests. The individual confidence intervals from Bonferroni t-tests are uniformly shorter than S-intervals implied by the F-test. Power curves are constructed for a few specific alternative hypotheses as functions of the correlation between regressors for the special case of two hypotheses and two regressors. The power of the two procedures is similar when the correlation is small. For highly correlated regressors, however, the power of the Bonferroni method is generally inferior. Thus, if regressors can be controlled to be uncorrelated, the Bonferroni method is clearly superior; otherwise neither method dominates. [ABSTRACT FROM AUTHOR]

In experimental designs with randomized blocks, estimates of the inter-block variance often have few degrees of freedom since they depend only on the block averages (the inter-block information). These degrees of freedom are further reduced by the number of parameters appearing in the inter-block information. However, some parameters also appear in the intra-block information, and thus allow two independent estimators. The difference of these two estimators provides additional information about the inter-block variance. This paper shows how this extra information may be recovered to improve tests of hypotheses concerning inter-block parameters. [ABSTRACT FROM AUTHOR]

The combined use of direct estimation (in saturated models) and indirect testing (of unsaturated models), proposed in [7, 8] for the analysis of a given multidimensional contingency table, is extended here to the analysis of a set of T multidimensional tables (T is greater than or equal to 2). For this set of tables, a three-fold classification of models is introduced: (1)models that assume "complete homogeneity" among the tables; (2) models that allow "complete heterogeneity" among the tables; and (3) models that allow "partial heterogeneity." Stepwise procedures proposed in [8] for model selection are extended. "Guided" and "unguided" selection methods and "multidirectional" methods are introduced. For illustrative purposes, a set of two 4-way tables is analyzed. [ABSTRACT FROM AUTHOR]

In paired comparisons of a treatment and control, it frequently happens that the two members of each pair can be classified into distinguishable types. The completely randomized design, which assigns the members of each pair at random to treatment and control, then may by chance assign the treatment primarily to subjects of one type and thereby confound treatment and type. This difficulty can be avoided by restricting the randomization. We find that such restriction is desirable by analyzing several of the standard tests (Wilcoxon, t, and tests for dichotomous response) for efficiency and deficiency. [ABSTRACT FROM AUTHOR]

This study is concerned with deriving the properties of the preliminary test estimator for the general linear normal regression model, ascertaining the characteristics of the risk functions over the parameter space, and determining the conditions necessary for the risk of this estimator to exceed or be less than the conventional one under squared error loss. A test procedure and the problem of choosing an optimal level of significance for the test are discussed. Some theorems and lemmas used in evaluating the risk and some properties of functions of the non-central F distribution are developed in the appendices. [ABSTRACT FROM AUTHOR]

Use of the Bonferroni inequality for controlling family Type I error rate is discussed and new tables of t and chi square are provided. The tabled percentage points provide the capability for unequal Type I error allocation over a family of statistical tests. Use of these tables is illustrated. [ABSTRACT FROM AUTHOR]

This study concerns the relative efficiency of the least-squares estimator of the mean of a covariance stationary sequence that has a constant mean and an autoregressive representation of order p. Our results augment those in [2]. The results in Section 1 include for any positive integer p the variance of the LSE, the BLUE in terms of autoregressive coefficients, the variance of the BLUE and the range of applications of Ylvisaker's theorem [7] regarding a lower bound on relative efficiency. For second-order schemes it also includes bounds on parameter values for relative efficiency is greater than or equal to 0.80. Graphical results for the first-order scheme in Section 2 add to the detailed study made in [2] and imply that the relative desirability of the LSE should be based on variance as well as relative efficiency considerations. [ABSTRACT FROM AUTHOR]

Investigates the power function of the F-test in analysis of variance under non-normal situations. Background on attempts to investigate the power function of F-tests; Numerical values of the power; Effect of non-normality on the power of the F-test. [ABSTRACT FROM AUTHOR]

Suppose that independent samples are available from two normal populations with means mu[sub 1] and mu[sub 2] and known variances. It is desired to test H: mu[sub 1] = 0. A possible preliminary test procedure in this case consists of first testing mu[sub 1] = mu[sub 2] and then testing H using either the sample mean from the first population or the pooled sample mean depending on the outcome of the test regarding equality of mu[sub 1] and mu[sub 2]. This procedure is shown to be biased. A power comparison of this procedure with the generalized likelihood ratio test of H indicates that the latter test is more powerful in a fairly extensive parametric region. It is noted that biasedness is frequently encountered in preliminary test procedures. In light of the example, it is recommended that power comparisons be made between such procedures and any available unbiased tests to ascertain whether extensive power loss regions exist or not. [ABSTRACT FROM AUTHOR]

The principle of minimum discrimination information estimation is described and used to generate estimates for tests of hypotheses concerning various interactions and effects in the analysis of multidimensional contingency tables. All classical hypotheses for contingency tables can be generated by the use of this principle when certain marginals are considered as fixed. Analysis of information tables are given for a four-way contingency table. [ABSTRACT FROM AUTHOR]

A test is scale and location invariant when the decision to accept or reject is not affected by multiplying each data point by the same positive constant or by adding the same constant to each data point. Two well-known statistics are shown to be test statistics of certain most powerful scale and location invariant tests: the ratio of range to standard deviation for normality versus uniformity, and the ratio of minimum deviation to standard deviation for normality versus exponentiality. The most powerful scale and origin invariant test of uniformity versus exponentiality is also considered. [ABSTRACT FROM AUTHOR]

This article investigates empirical evidence on the structure of price expectations in the United States during the post-Korean War period. The study utilizes semiannual data which describe price expectations for six months and twelve months ahead. The objective is to use these two sets of data to test some of the well-known expectational hypotheses. Incidental to this we determine whether expectations satisfy the rationality hypothesis, and we briefly consider the accuracy of the predictions. [ABSTRACT FROM AUTHOR]

Under a reasonable hypothesis about the distribution of trade shipments among a group of K countries, or of non-self-directed acts among any group of K individuals, the likelihood of obtaining a given table of observed frequencies n[sub ij] is [Multiple line equation(s) cannot be represented in ASCII text] where n[sub I]=n[sub i1] + ... + n[sub iK] and m[sub I] = n[sub 1i] + ... + n[sub K I] are the total number of shipments made and received, respectively, by country I, N is the total number of shipments altogether, and Pi and Q[sub I] are the theoretical tendencies of country I to ship and to receive shipment respectively and satisfy SIGMA P[sub I] = SIGMA Q[sub I] = 1. In the attempt to test the correctness of the hypothesis, a critical problem is that of finding the maximum-likelihood estimates of the P's and Q's, that is, the values making L as large as possible. We give all solutions and show that, for the usual table of observations, the solution is unique and amounts to finding the value of a for which K-2 -R[sub 1] ... - R[sub K] is 0 - [Multiple line equation(s) cannot be represented in ASCII text] [ABSTRACT FROM AUTHOR]

Cronholm and Revusky [4] have proposed a rank test based on independent subexperiments for the comparison of a treatment effect with a control when the treatment is destructive but the control produces at most a transient effect. This article provides a thorough study of its performance under two sampling situations. With the aid of a computer, the exact small sample power of the test is evaluated for some important alternatives. A modification of the test is proposed for the situation where numerical measurements are available rather than merely the ranks within each stage. The modification is found to substantially improve the power. Both tests are shown to be unbiased. Finally, the Pitman asymptotic efficiencies are obtained and comparisons are made with the appropriate Wilcoxon tests. [ABSTRACT FROM AUTHOR]

For Bernoulli random variables sequential tests of the simple hypotheses p = p[SUB o] vs. p = p[sub 1] are considered. In particular a comparison is made between the performance of sequential probability ratio tests (SPRTs) and curtailed single sampling plans (CSSPs), when both tests have the same error probabilities. In [1] it is shown that the CSSP has a strong optimality property. Nevertheless, if one admits randomized SPRTs, the latter are better. Numerical examples are considered in detail for p[sub o] = 1/2 and the CSSP with n = 2. The method of finding the randomized SPRTs with the correct error probabilities is indicated. These rules have positive probability of deciding without taking any observations. [ABSTRACT FROM AUTHOR]

This study is concerned with the extension of the Bradley-Terry model for paired comparisons to situations which allow an expression of no preference. A new model is developed and its performance compared with a model proposed by Rao and Kupper. The maximum likelihood estimates of the parameters are found using an iterative procedure which, under a weak assumption, converges monotonically to the solution of the likelihood equations. It is noted that for a balanced paired comparison experiment the ranking obtained from the maximum likelihood estimates agrees with that obtained from a scoring system which allots two points for a win, one for a tie and zero for a loss. The likelihood ratio test of the hypothesis of equal preferences is shown to have the same asymptotic efficiency as that for the Rao-Kupper model. Two examples are presented, one of which introduces a set of data for an unbalanced paired comparison experiment. Initial applications of the test of goodness of fit suggest that the proposed model yields a reasonable representation of actual experimentation. [ABSTRACT FROM AUTHOR]

There is not much in the literature on tolerance limits for discrete distributions, since generally one will find it difficult to derive explicit expressions for these limits. Distribution-free tolerance limits for discrete cases were discussed. If the form of the distribution function is known, one should try to determine the formula of the tolerance limits for the particular family of distributions to which it belongs. Otherwise, an important information is not utilized. In applications for sampling inspection reliability, inventory control, the observed random variables are very often assumed to have a binomial, negative-binomial or Poisson distribution. In these cases one can obtain explicit expressions for tolerance limits, which have desirable optimum properties. In the present article we derive uniformly most accurate upper tolerance limits for families of distributions with the monotone likelihood ratio property. For these families there exist uniformly most accurate upper confidence limits. Furthermore, the distribution functions which belong to monotone likelihood ratio families are stochastically ordered. Thus, monotone relationship exists in these cases between tolerance limits and confidence limits. This relationship is utilized to derive the formulae of uniformly most accurate tolerance limits. The general theory is presented in Section 2. Section 3 is devoted to uniformly most accurate upper confidence limits.

Natural test statistics for the hypothesis that an equation is overidentified have been developed by Anderson and Rubin and by Basmann. If the disturbances are jointly normal, serially uncorrelated, and small, both the above overidentification test statistics have the Fisher variance-ratio distribution asymptotically as the variance of the error terms gets small. This gives an analytic explanation of Monte Carlo results of Basmann. The results given apply to linear models in which predetermined variables are exogenous. [ABSTRACT FROM AUTHOR]

A table is presented containing critical points for the non-central F with non-centrality of one-half for type one error .05, .10, .25 and .50 and numerator and denominator d.f. including 1-30, 40, 60, 120, 200,400, 1000. The critical points may be used to test the hypothesis that aset of exact linear restrictions imposed on the parameters of a linear model yield estimators that are better in generalized MSE than the unrestricted, o.l.s., estimators. Tabulation is discussed and some examples are presented on how the tables may be used. [ABSTRACT FROM AUTHOR]

A double sample test for a hypothesis concerning the mean mu of a normal random variable with unknown variance sigma[sup 2] is presented in this note. This test is an alternative to the usual single sample t test and is made by taking samples at two stages. After the first sample has been observed the hypothesis is rejected, accepted or a second sample is taken. This double sample test is a modification of a double sample test suggested by D. Owen. However, for the test being suggested here if the second sample is taken the information from the first sample is used in making the final decision. A table is included which is a tabulation of the rejection and acceptance points for various sample sizes and alpha = .05 and alpha = .01. Additional tables are included which give some power values and expected sample sizes for the new test and Owen's test. These power values are compared with power values of various single sample t tests. Conclusions about desirable sample sizes are presented for both the new test and Owen's test. [ABSTRACT FROM AUTHOR]

A new class of tests is presented for testing the hypothesis of no second order interaction in three-way contingency tables. This hypothesis is expressed first in the form of linear contrasts in logarithms of the parameters of a multinomial experiment and the test is based on the general theory of locally asymptotically most stringent tests. Specific test criteria are then given for testing no second order interaction in 2 X2 X 2, 2 x 2 x K and I x J x K contingency tables. Two examples are discussed. [ABSTRACT FROM AUTHOR]

Results are given of an empirical sampling study of the sensitivities of nine statistical procedures for evaluating the normality of a complete sample. The nine statistics are W (Shapiro and Wilk, 1965), square root of b[sub I] (standard third moment), b[sub 2] (standard fourth moment), KS (Kolmogorov-Smirnov), CM (Cramer-Ven Mises), WCM (weighted CM), D (modified KS), CS (chi-squared) and u (Studentized range). Forty-five alternative distributions in twelve families and five sample sizes were studied. Results are included on the comparison of the statistical procedures in relation to groupings of the alternative distributions, on means and variances of the statistics under the various alternatives, on dependence of sensitivities on sample size, on approach to normality as measured by the W statistic within some classes of distribution, and on the effect of misspecification of parameters on the performance of the simple hypothesis test statistics. The general findings include: (I) The W statistic provides a generally superior omnibus measure of non-normality; (ii) the distance tests (KS, CM, WCM, D) are typically very insensitive; (iii) the u statistic is excellent against symmetric, especially short-tailed, distributions but has virtually no sensitivity to asymmetry; (iv) a combination of both square root of b[sub 1] and b[sub 2] usually provides a sensitive judgment but even their combined performance is usually dominated by W; (v) with sensitive procedures, good indication of extreme non-normality (e.g., the exponential distribution) can be... [ABSTRACT FROM AUTHOR]

The article discusses the analysis of cross-classified data. There are two different kinds of contingency tables, that is, truncated contingency tables in which the entries in some of the cells of the table are omitted from the analysis and the more usual contingency tables in which none of the cells of the table are omitted. In the usual non-truncated contingency table, one need to calculate statistician Karl Pearson's chi-square statistic for testing the null hypothesis of independence in the table, using the appropriate number of degrees of freedom. The concept of "quasi-independence" is useful in analyzing tables for which some of the cells are truncated and also in analyzing the more usual kinds of cross-classification tables since it leads to methods that focus attention in turn on various subsets of the entire table, making possible a more detailed analysis of the association between the row and column classifications in the table. For the non-truncated cross-classification table, if the rows and columns are not independent, it will often be of interest to estimate what are the kinds of interaction or association present in the table and what are their magnitudes.

Several writers have studied the problem of obtaining confidence bands for a straight line regression under the requirement that the bands be at a specified confidence level, (1 - α) say, for values of the independent variable restricted to a pre-specified closed interval. The bands studied by other writers have been either bands of equal width throughout the interval or trapezoidal bands. This paper studies confidence bands of the classical hyperbolic type under the restriction noted above. Relevant distribution results differ according to whether the pre-specified interval on the independent variable is symmetric or asymmetric about the mean of the independent variable values in the experiment. In the former case the problem under study, is shown to be equivalent to a problem studied by Halperin et al (JASA, Sept. 1967) for which distribution theory and tables are already available. The symmetrical case is generalized to obtain confidence bands in multiple linear regression at level (1 - α) for an ellipsoidal region on the independent variables centered at the point of means of the independent variable values used in the experiment. For straight line regression some numerical comparisons are made with bands of the other types mentioned above. These comparisons suggest that the bands proposed in this paper are uniformly superior to bands of equal width in the sense of having smaller average width; the proposed bands appear to be superior to trapezoidal bands in the same sense for intervals of practical interest, i.e. within or reasonably close to the range on the independent variables used in the experiment. [ABSTRACT FROM AUTHOR]

The first part of this note is devoted to some of the simpler series representations of the doubly non-central t-distribution. In the second part, the computational considerations arising from the series representations are discussed. A table showing some of the results of the method is presented. [ABSTRACT FROM AUTHOR]

The topic of the shrinkage of an estimator for the mean of a normal distribution towards a point ([3] Thompson) is extended to shrinkage to an interval centered at μ = 0. An effort is made to keep the form of the modified estimator relatively simple, while obtaining more flexibility than is possessed by the sample mean x. The family of estimators advocated is: μ = x + k σ2x/4θ ln [(x - θ )2 + kσ 2x/ (x + &#x03B8)2 + k σ 2x. Curves of MSE/σ2x (where MSE is mean square error) versus |μ/σx| are given for a number of values of θ and k. A table is given by the use of which k and θ may be selected on the basis of the minimum value of MSE/σ2/x, the width of the interval for which MSE/σ2/x < 1, the maximum value of MSE/σ2/x, and the value of [μ/σx| for which MSE/σ2/x is a maximum. [ABSTRACT FROM AUTHOR]

This note gives a systematic method of computing the probability P[sub n](beta) of the event [Multiple line equation(s) cannot be represented in ASCII], where F[sub n] and F are the empirical and theoretical cumulative distribution functions and beta is a function on the interval (0, 1). The result is used to determine not only the significance level but also the exact power for an appropriate testing situation. When beta is a linear function, P[sub n](beta) may represent the probability distribution of the one-sided Kolmogorov-Smirnov statistic D[sup +, sub n]=sup[sub -Infinity < x < Infinity] [F[sub n(x) - F(x)]. Numerical examples are given in section 2, and they show that the method is extensively useful for practical applications. [ABSTRACT FROM AUTHOR]

Tests are developed concerning the mean of the three-parameter Weibull distribution with the variance known and with the variance unknown, and concerning the variance with the mean unknown. The shape parameter is considered known. These tests are compared with tests developed by Hatter and Dubey based on the usual statistics for the normal case. A two-sample test for difference in means and for difference in reliability is also discussed. For the most part the tests are simple yet provide improvements over existing methods. [ABSTRACT FROM AUTHOR]

Some optimum nonparametric procedures for estimating and testing contrasts in two-way layouts are proposed and studied. These procedures are based on the Chernoff-Savage [4] type of rank order statistics which include the Wilcoxon and normal scores statistics among others. In the first three sections, the properties of the proposed point estimators of contrasts, such as symmetry, invariance and asymptotic normality, are studied and their asymptotic relative efficiencies with respect to the corresponding least-squares estimators are obtained. In particular, it is shown that the procedures based on the normal scores statistics are asymptotically at least as efficient as the corresponding procedures based on the method of least squares, whatever be parent cumulative distribution functions. In sections 4 and 5, the corresponding problems of testing and confidence intervals are discussed, and generalized to two-way layouts with several observations per cell. [ABSTRACT FROM AUTHOR]